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Relative algebroids and Cartan realization problems

Rui Loja Fernandes, Wilmer Smilde

TL;DR

This work introduces relative algebroids, unifying Lie algebroids and (formal) PDEs by encoding a degree-1 derivation relative to a submersion or foliation. It develops a formalism of relative derivations, tableaux, and prolongations; constructs a universal relative algebroid; and extends Spencer theory to derive obstructions (torsion and curvature) to realizations and formal integrability. The framework shows that PDEs can be recast as relative algebroids, with prolongation theory aligning with Goldschmidt’s formal integrability results, and it provides mechanisms for reduction and natural constructions that preserve realizations. It also connects Bryant’s equations to this theory, outlines natural examples, and points to downstream applications in Cartan’s realization problem and geometric classifications.

Abstract

We develop a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations.

Relative algebroids and Cartan realization problems

TL;DR

This work introduces relative algebroids, unifying Lie algebroids and (formal) PDEs by encoding a degree-1 derivation relative to a submersion or foliation. It develops a formalism of relative derivations, tableaux, and prolongations; constructs a universal relative algebroid; and extends Spencer theory to derive obstructions (torsion and curvature) to realizations and formal integrability. The framework shows that PDEs can be recast as relative algebroids, with prolongation theory aligning with Goldschmidt’s formal integrability results, and it provides mechanisms for reduction and natural constructions that preserve realizations. It also connects Bryant’s equations to this theory, outlines natural examples, and points to downstream applications in Cartan’s realization problem and geometric classifications.

Abstract

We develop a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations.

Paper Structure

This paper contains 40 sections, 49 theorems, 293 equations.

Table of Contents

  1. Relative derivations
  2. Derivations and brackets
  3. Relative derivations
  4. Relative brackets
  5. Brackets and derivations relative to foliations
  6. Foliated flat vector bundles
  7. Derivations relative to a foliation
  8. The structure of relative derivations
  9. Morphisms and extensions of relative derivations
  10. Tableaux of derivations
  11. Definition of tableau of derivations
  12. Spencer cohomology
  13. Cartan characters and Cartan's test
  14. Symbol exact sequences of tableaux and involutivity
  15. Relative algebroids
  16. ...and 25 more sections

Key Result

Theorem 1

Let $(A,p,{\mathrm{D}})$ be a relative algebroid with tableau of derivations $\tau$. Suppose that: Then $(A,p,{\mathrm{D}})$ is formally integrable.

Theorems & Definitions (180)

  • Example 1: Space forms
  • Example 2: Surfaces with $|\nabla K | = 1$, Bryant2014
  • Theorem 1: Theorem \ref{['thm:GoldschmidtsFormalIntegrabilityCriterion']}
  • Theorem 2: Bryant-Cartan, Theorem \ref{['thm:CartanBryantExistence']}
  • Theorem 3: Theorem \ref{['thm:PDEsAndRelativeAlgebroids']}
  • Proposition 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.4
  • Lemma 1.5
  • ...and 170 more